## Sinusoidal and Exponential Sequences, Periodicity of Sequences

Continuing our discussion on discrete-time sequences, we now come to define exponential and sinusoidal sequences. The general formula for a discrete-time exponential sequence is as follows:

x[n] = Aα^n.

This exponential behaves differently according to the value of α. If the sequence starts at n=0, the formula is as follows:

x[n] = Aα^n * u[n].

If α is a complex number, the exponential function exhibits newer characteristics. The envelope of the exponential is |α|. If |α| < 1, the system is decaying. If |α|> 1, the system is growing.

When α is complex, the sequence may be analyzed as follows, using the definition of Euler’s formula to express a complex relationship as a magnitude and phase difference.

Where ω0 is the frequency and φ is the phase, for n number of samples, a complex exponential sequence of form Ae^jw0n may be considered as a sinusoidal sequence for a set of frequencies in an interval of 2π.

A sinusoidal sequence is defined as follows:

x[n] = Acos(ω0*n + φ), for all n, and A, φ are real constants.

Periodicity for discrete-time signals means that the sequence will repeat itself for a certain delay, N.

x[n] = x[n+N] : system is periodic.

t = (-5:1:15)’;

impulse = t==0;

unitstep = t>=0;

Alpha1 = -0.5;

Alpha2 = 0.5;

Alpha3 = 2.5;

Alpha4 = -2.5;

cAlpha1 = -0.5 – 0.5i;

cAlpha2 = 0.5 + 0.5i;

cAlpha3 = 2.5 -2.5i;

cAlpha4 = -2.5 + 2.5i;

A = 1;Exp1 = A.*unitstep.*Alpha1.^t;

Exp2 = A.*unitstep.*Alpha2.^t;

Exp3 = A.*unitstep.*Alpha3.^t;

Exp4 = A.*unitstep.*Alpha4.^t;cExp1 = A.*unitstep.*cAlpha1.^t;

cExp2 = A.*unitstep.*cAlpha2.^t;

cExp3 = A.*unitstep.*cAlpha3.^t;

cExp4 = A.*unitstep.*cAlpha4.^t;%%

figure(1)

subplot(2,1,1)

stem(t, impulse)

xlabel(‘x’)

ylabel(‘y’)

title(‘Impulse’)subplot(2,1,2)

stem(t, unitstep)

xlabel(‘x’)

ylabel(‘y’)

title(‘Unit Step’)

%%

figure(2)

subplot(2,2,1)

stem(t, cExp1)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = -0.5 – 0.5i’)subplot(2,2,2)

stem(t, cExp2)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = 0.5 + 0.5i’)subplot(2,2,3)

stem(t, cExp3)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = 2.5 -2.5i’)subplot(2,2,4)

stem(t, cExp4)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = -2.5 + 2.5i’)

%%

figure(3)

subplot(2,2,1)

stem(t, Exp1)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = -0.5’)subplot(2,2,2)

stem(t, Exp2)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = 0.5’)subplot(2,2,3)

stem(t, Exp3)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = 2.5’)subplot(2,2,4)

stem(t, Exp4)

xlabel(‘n’)

ylabel(‘x[n]’)

title(‘Exponential: alpha = -2.5’)

## Reply